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To PI or not to PI challenge. Stern-Brocot

  1. May 7, 2010 #1
    Dear readers,


    At my webpage:
    http://domingogomez.web.officelive.com/brocotfraction.aspx [Broken]

    I have posed a challenge entitled "To PI or not to PI", it deals with
    a generalized continued fraction (bifurcated fraction) whose
    coeficients are all the Stern-Brocot fractions, that is, all the positive
    rational numbers in their reduced forms.
    This very special fractal fraction converges to a very specific value
    close to PI.
    I hope you find of some interest this challenge.

    Best regards,

    Domingo Gomez Morin
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. May 7, 2010 #2
    Code (Text):
    /* PARI/GP */
    sbf(a,b,c,d,h)={
        h--;
        if(h<=0, return((a+c)/(b+d)));
        return( (a+c)/(b+d)+sbf(a+c,b+d,c,d,h)/sbf(a,b,a+c,b+d,h));
    }

    /* faster */
    sbF(a,b,c,d,h)={
        h--;
        if(h<=0, return( 0.0 + (a+c)/(b+d)));
        return( (a+c)/(b+d)+sbF(a+c,b+d,c,d,h)/sbF(a,b,a+c,b+d,h));
    }
    sbf(0,1,1,0,3) = 13/5 = 2.6

    for(i=0,18,print(i" "sbf(0,1,1,0,i)+.))
    0 1.0000000000000000000000000000000000000000000000000
    1 1.0000000000000000000000000000000000000000000000000
    2 5.0000000000000000000000000000000000000000000000000
    3 2.6000000000000000000000000000000000000000000000000
    4 3.4626379400945874934314240672622175512348922753547
    5 2.9388088765046610642144686035478929406022355808575
    6 3.2735185568492236282348448216538099507168494439501
    7 3.0347992453939067319279281866337440790873147702144
    8 3.2099684860688422980323591345066881944750995102337
    9 3.0738506312040716737872747524887840798894383527984
    10 3.1814697094275065992958010264534251536404002098575
    11 3.0934945502880832477021949766849855361715888355938
    12 3.1663219592217772133098797759764429309295906541622
    13 3.1047408775416900905987339929906019611661463991880
    14 3.1573142239629423684606802771536687757026460862718
    15 3.1117718703279799014367140110065699129312803115134
    16 3.1515201645394594475594374322542040402712357473677
    17 3.1164582339359882762699683957496012711553724723406
    18 3.1475716978017248990506977242760373644809795171547
    number of terms: 2^(i+1)-1
    2^19-1=524287

    C program, doubles (34 359 738 367 terms)
    Code (Text):
      0   1.00000000000000000
      1   1.00000000000000000
      2   5.00000000000000000
      3   2.60000000000000010
      4   3.46263794009458700
      5   2.93880887650466120
      6   3.27351855684922330
      7   3.03479924539390660
      8   3.20996848606884240
      9   3.07385063120407190
     10   3.18146970942750640
     11   3.09349455028808280
     12   3.16632195922177710
     13   3.10474087754169000
     14   3.15731422396294280
     15   3.11177187032798040
     16   3.15152016453945950
     17   3.11645823393598760
     18   3.14757169780172540
     19   3.11973774954432280
     20   3.14475925418342990
     21   3.12212230271433850
     22   3.14268465110109930
     23   3.12391046202905590
     24   3.14111024668793840
     25   3.12528588642403580
     26   3.13988701664436620
     27   3.12636658593696290
     28   3.13891762963651200
     29   3.12723119902446500
     30   3.13813631476721520
     31   3.12793376938022140
     32   3.13749732276145020
     33   3.12851243406977100
     34   3.13696802810121510
    Code (Text):
    sumalt_partial(p, n=#p)={ local(b,c,s);
        b=2^(2*n-1);
        c=b;
        s=0;
        forstep(k=n-1,0,-1,
            s+= b*p[k+1];
            b*= ((2*k+1)*(k+1))/(2*(n+k)*(n-k));
            c+= b;
        );
        s/c
    }
    t[] computed in C
    Code (Text):
    PARI/GP>for(i=1,#t,print(i"   "altsum_partial(t,i)))
    1   0.66666666666
    2   0.94117647058
    3   2.28282828282
    4   3.12790294627
    5   3.31575100366
    6   3.27005398107
    7   3.20530124624
    8   3.16657805849
    9   3.14828078568
    10   3.1403458694
    11   3.1368920474
    12   3.1352752842
    13   3.1344294778
    14   3.1339363396
    15   3.1336247926
    16   3.1334175662
    17   3.1332751260
    18   3.1331749115
    19   3.1331030537
    20   3.1330506392
    21   3.1330117843
    22   3.1329825333
    23   3.1329601875
    24   3.1329428795
    25   3.1329293002
    26   3.1329185187
    27   3.1329098646
    28   3.1329028481
    29   3.1328971071
    30   3.1328923702
    31   3.1328884319
    32   3.1328851345
    33   3.1328823558
    34   3.1328800237
    New, fundamental constant 3.13288....... :smile:
     
    Last edited: May 7, 2010
  4. May 7, 2010 #3
    Xitami says:
    >New, fundamental constant 3.13288.......

    Well that seems to be the answer to Shakespeare's question: Not to PI, just close.

    That was fast Xitami ¡¡¡, thanks a lot, let's see what others have to say about
    your result.
    Please let me know your name (by email) so I can include it for the records in my webpage,
    or let me know if you prefer to use "Xitami".

    I think this new constant might be called: "Universal Rational-Mean Constant"


    Cheers,

    Domingo Gomez
     
  5. May 7, 2010 #4
    When using Farey fractions instead of Stern-Brocot's then the new constant is approx.:

    1.226...

    which is not "Universal" :smile: because it does not embrace all the positive rational numbers in their reduced forms but just those from 0 to 1.

    So this one could be called: "Farey Rational Constant"

    Could the reader find a relation between both of them?
     
  6. May 7, 2010 #5
    alternting variant
    Code (Text):

            2/1 + ---
    1/2 -  ----------
            1/2 + ---

      0      1.00000000000000000
      1      1.00000000000000000
      2     -3.00000000000000000
      3      1.00000000000000000
      4     -6.91232876712328980
      5      0.14328134217554114
      6     -3.87884017589414440
      7      0.69975974137673203
      8     -8.44206995530308470
      9     -0.76694421386509692
     10     -3.65503802208626280
     11      2.35578956476438380
     12      6.23413307523992440
     13    -19.28387468639876800
     14      2.89341005066847280
     15      0.32136221024384604
     16      1.16624268747289440
     17      0.80821242069640231
     18      1.04301471087375220
     19      0.95481756005451746
     20      1.01034312761793440
     21      0.98854958616575817
     22      1.00233394417308630
     23      0.99836436971475051
     24      1.00006518857518570
     25      0.99998636872615365
     26      1.00000017834156240
     27      0.99999838543925079
     28      1.00000000642554520
     29      0.98723877363144263
     30      1.18727245568190480
     31     15.64720874911006400
     32      0.99427527762970824
    strange
    Regards, Xitami.
     
  7. May 7, 2010 #6
    1.226... ?
    pari/gp> for(i=1,20,print(sbf(0,1,1,1.,i)))
    0.500000000000000000
    2.500000000000000000
    1.491379310344827586
    1.913773073178070399
    1.627260322338834205
    1.819867311604092935
    1.675122681756296973
    1.785265731369331449
    1.697124869319332277
    1.768470830315575050
    1.709007940366553484
    1.759014903604435405
    1.716146298615272456
    1.753157353718325420
    1.720768312861207621
    1.749275249557767505
    1.723932145152701193
    1.746568856199571096
    1.726192767930516595
    1.744606292952420387
     
  8. May 7, 2010 #7
    Yes Xitami, it depends on the colocation of the initial fractions (0/1, 1/1)
    or (1/1, 0/1) (for the fractal fraction it means: above or below respectively)

    That is, starting with: 1/2+ (2/3)/(1/3)...
    or
    starting with 1/2+ (1/3)/(2/3) ...

    got it? So both constants are right: 1.226... and 1.7...

    The same applies for the :smile: Shakespeare's constant 3.13288....
    please try by using (1/0, 0/1) instead of (0/1, 1/0) for that case, and you will see...

    All of them are new constants (My conjecture is that they are Transcendental Numbers)

    Interesting, isn't it?
     
  9. May 7, 2010 #8
    just a typo: in my last mesasge I said "colocation" and I meant "LOCATION"
     
  10. May 8, 2010 #9
    Yes
    for i = 1..32
    Code (Text):
          0  oo              oo  0             0  1              1  0              1  oo             oo  1
    ------------------------------------------------------------------------------------------------------------
        0       1           1      0         0      1          1      0          1       1         1       1
        -       -           -      -         -      -          -      -          -       -         -       -
        1       0           0      1         1      1          1      1          1       0         0       1
    ------------------------------------------------------------------------------------------------------------[
    [1.00000000000000, 1.00000000000000, 0.50000000000000, 0.50000000000000, 2.00000000000000, 2.00000000000000],
    [5.00000000000000, 1.25000000000000, 2.50000000000000, 1.00000000000000, 4.00000000000000, 2.50000000000000],
    [2.60000000000000, 1.40000000000000, 1.49137931034483, 1.15340909090909, 3.67272727272727, 2.63448275862069],
    [3.46263794009459, 1.43781235126130, 1.91377307317807, 1.17893945899528, 3.71044022189325, 2.63542013102534],
    [2.93880887650466, 1.44734402880067, 1.62726032233883, 1.19791711626815, 3.69960653966179, 2.63610535624420],
    [3.27351855684922, 1.45442687388447, 1.81986731160409, 1.20683409722323, 3.70306463236905, 2.63646628086641],
    [3.03479924539391, 1.45774683559641, 1.67512268175630, 1.21264226915310, 3.70196833698054, 2.63648082287349],
    [3.20996848606884, 1.45994731258142, 1.78526573136933, 1.21650578913913, 3.70237446386729, 2.63650648379447],
    [3.07385063120407, 1.46140822964650, 1.69712486931933, 1.21912169269838, 3.70222649553624, 2.63650960649757],
    [3.18146970942751, 1.46239986749675, 1.76847083031558, 1.22107287639257, 3.70228404560910, 2.63651175275555],
    [3.09349455028808, 1.46313955366077, 1.70900794036655, 1.22249580904769, 3.70226142970045, 2.63651227368401],
    [3.16632195922178, 1.46367916480036, 1.75901490360444, 1.22360568989495, 3.70227057182131, 2.63651250401425],
    [3.10474087754169, 1.46410008980877, 1.71614629861527, 1.22446640266835, 3.70226682040408, 2.63651258450038],
    [3.15731422396294, 1.46442653445570, 1.75315735371832, 1.22515692996179, 3.70226838384100, 2.63651261462383],
    [3.11177187032798, 1.46468843849495, 1.72076831286121, 1.22571570267877, 3.70226772362143, 2.63651262714428],
    [3.15152016453946, 1.46490037258285, 1.74927524955777, 1.22617487185787, 3.70226800534697, 2.63651263166333],
    [3.11645823393599, 1.46507452956305, 1.72393214515270, 1.22655727672556, 3.70226788386056, 2.63651263367106],
    [3.14757169780173, 1.46521957113390, 1.74656885619957, 1.22687824862140, 3.70226793666468, 2.63651263441218],
    [3.11973774954432, 1.46534131208324, 1.72619276793052, 1.22715108699814, 3.70226791352349, 2.63651263474630],
    [3.14475925418343, 1.46544479659443, 1.74460629295242, 1.22738428973423, 3.70226792373012, 2.63651263487502],
    [3.12212230271434, 1.46553324778298, 1.72786411537339, 1.22758564599825, 3.70226791919895, 2.63651263493274],
    [3.14268465110110, 1.46560961997042, 1.74313747561861, 1.22776037989534, 3.70226792122129, 2.63651263495598],
    [3.12391046202906, 1.46567589459545, 1.72913465153928, 1.22791317933125, 3.70226792031393, 2.63651263496631],
    [3.14111024668794, 1.46573384972492, 1.74200936653856, 1.22804745180117, 3.70226792072287, 2.63651263497062],
    [3.12528588642404, 1.46578477778274, 1.73012308207908, 1.22816613189901, 3.70226792053776, 2.63651263497252],
    [3.13988701664437, 1.46582979182719, 1.74112401178017, 1.22827151794382, 3.70226792062188, 2.63651263497334],
    [3.12636658593696, 1.46586976358497, 1.73090719777399, 1.22836552708201, 3.70226792058352, 2.63651263497370],
    [3.13891762963651, 1.46590542020834, 1.74041633194309, 1.22844974450798, 3.70226792060107, 2.63651263497386],
    [3.12723119902447, 1.46593736294390, 1.73153970353667, 1.22852547356149, 3.70226792059301, 2.63651263497393],
    [3.13813631476722, 1.46596608613394, 1.73984170648085, 1.22859382814243, 3.70226792059672, 2.63651263497396],
    [3.12793376938022, 1.46599201226834, 1.73205733694769, 1.22865572486824, 3.70226792059501, 2.63651263497398],
    [3.13749732276145, 1.46601548900992, 1.73936870714773, 1.22871196056974, 3.70226792059580, 2.63651263497398]];
    ----------------------------------------------------------------------------------------------------------------
     3.13286308929902, 1.46633409868672, 1.73582085736194, 1.22950308242259, 3.70226792059555, 2.63651263497399 Wynn
     3.13288122714498, 1.46562502661973, 1.73583922196094, 1.22779888812588, 3.70226792059623, 2.63651263487680 SumAlt
     3.13294527734985, 1.46483896033168, 1.73589528981583, 1.22606704915966, 3.70226791233679, 2.63651256289041 Aver
     
     
  11. May 8, 2010 #10
    Great job Xitami ¡¡¡
     
  12. May 8, 2010 #11
    Well I made a lookup at Plouffe's inverter:
    http://pi.lacim.uqam.ca/

    and, of course, we need better approximations to grasp a bit of the essence
    of all this.

    Please Xitami take a lookup of all those values at Plouffe's inverter.
     
  13. May 10, 2010 #12
    [URL]http://fotoo.pl/zdjecia/files/2010-05/a512eda9.png[/URL]
     
    Last edited by a moderator: Apr 25, 2017
  14. May 15, 2010 #13
    Indeed, many thanks Xitami.
    It would be great if you could make a brief explanation on the amazing graph you
    have posted, thanks again.:approve:

    On the other hand, I would like to make the following question:

    Does anyone here know about any book or paper describing the ARITHMETIC MEAN
    of ALL RATIONAL NUMBERS?
    It seems to me that its value converges to 1.5, but I would like to know about any previous analysis on this in the math-literature.

    Based on such value 1.5, it is quite interesting to realize that the "EQUILIBRIUM POINT"
    of all rational numbers (between 0/1 and 1/0) is not 1, but 1.5. :confused:

    Isn't 1.5 interesting?


    ¿?¿?¿?¿?¿?¿?

    Next task for me: HARMONIC MEAN OF ALL RATIONAL NUMBERS
    :smile:
     
  15. May 15, 2010 #14
    >Next task for me: HARMONIC MEAN OF ALL RATIONAL NUMBERS

    Answer:
    Harmonic mean = 1/(Arithmetic Mean)
     
  16. May 16, 2010 #15
    about graph
    Code (Text):
    sbf(a,b,c,d,h){
        h--;
       
        plot( a+c , b+d );
       
        if(h<=0) return(a+c)/(b+d);
        return (a+c)/(b+d) + sbf(a+c,b+d,c,d,h) / sbf(a,b,a+c,b+d,h);
    }

    A(0,oo) A= 1+B[i-1] / C[i-1]
    B (1,oo)
    C (0,1)
    Code (Text):

     h |         A            |          B            |          C
    ---+----------------------+-----------------------+----------------------
    01 |       1.00           |  2.000000000000000000 |  0.500000000000000000
    02 | 5.000000000000000000 |  4.000000000000000000 |  2.500000000000000000
    03 | 2.600000000000000000 |  3.672727272727272400 |  1.491379310344827600
    04 | 3.462637940094587251 |  3.710440221893247100 |  1.913773073178070450
    05 | 2.938808876504660973 |  3.699606539661788800 |  1.627260322338834450
    06 | 3.273518556849223286 |  3.703064632369045900 |  1.819867311604092850
    07 | 3.034799245393906756 |  3.701968336980543000 |  1.675122681756296950
    08 | 3.209968486068842485 |  3.702374463867286600 |  1.785265731369331200
    09 | 3.073850631204071915 |  3.702226495536238700 |  1.697124869319332350
    10 | 3.181469709427506392 |  3.702284045609098000 |  1.768470830315575400
    11 | 3.093494550288082851 |  3.702261429700445900 |  1.709007940366553250
    12 | 3.166321959221777252 |  3.702270571821310700 |  1.759014903604435600
    13 | 3.104740877541689813 |  3.702266820404082700 |  1.716146298615272250
    14 | 3.157314223962942769 |  3.702268383840999500 |  1.753157353718324950
    15 | 3.111771870327980238 |  3.702267723621433400 |  1.720768312861207450
    16 | 3.151520164539459630 |  3.702268005346967900 |  1.749275249557767900
    17 | 3.116458233935987872 |  3.702267883860555800 |  1.723932145152700750
    18 | 3.147571697801725247 |  3.702267936664682700 |  1.746568856199571800
    19 | 3.119737749544323044 |  3.702267913523485300 |  1.726192767930516700
    20 | 3.144759254183429791 |  3.702267923730116400 |  1.744606292952420400
    21 | 3.122122302714338615 |  3.702267919198950300 |  1.727864115373394000
    22 | 3.142684651101099294 |  3.702267921221285500 |  1.743137475618610650
    23 | 3.123910462029055894 |  3.702267920313926800 |  1.729134651539278500
    24 | 3.141110246687938288 |  3.702267920722872600 |  1.742009366538558100
    25 | 3.125285886424035834 |  3.702267920537761500 |  1.730123082079081600
    26 | 3.139887016644366005 |  3.702267920621877400 |  1.741124011780174050
    27 | 3.126366585936963090 |  3.702267920583515600 |  1.730907197773988050
    28 | 3.138917629636511884 |  3.702267920601069600 |  1.740416331943094300
    29 | 3.127231199024464900 |  3.702267920593012000 |  1.731539703536669950
    30 | 3.138136314767215297 |  3.702267920596721100 |  1.739841706480854450
    31 | 3.127933769380221235 |  3.702267920595009100 |  1.732057336947687500
    32 | 3.137497322761450205 |  3.702267920595801300 |  1.739368707147728200
    33 | 3.128512434069770803 |  3.702267920595433600 |  1.732486341353947300
    34 | 3.136968028101215221 |  3.702267920595604600 |  1.738974675742910300
    35 | 3.128994730192923877 |  3.702267920595525100 |  1.732845868834206550
    36 | 3.136524654143805385 |  3.702267920595562000 |  1.738642940698007300
    37 | 3.129400944801941102 |  3.702267920595545100 |  1.733150161872439850
    38 | 3.136149539746592745 |  3.702267920595553100 |  1.738361013035048200
    39 | 3.129746291382634296 |           -           |          -
     
  17. May 16, 2010 #16
    Doskonały Xitami. Wielkie dzięki.

    Now I can see that 3.702267920595553 does not appear at Plouffe's Inverter database
    http://pi.lacim.uqam.ca/
    Now, it seems to me that it is very dificult to get enough decimal digits for the other cases.
    I wonder if it would be necesary to use a supercomputer, not just CUDA.
    Am I wrong?

    Now to the question on the arithmetic, harmonic and geometric mean of all rational numbers,
    I would really appreciate if someone could bring me any reference on papers or books relating this topic.

    The fact that the arithmetic mean of all rational numbers is 1.5 means (at least, from my point of view) that if all rational numbers were "entities" all of them having the same weight, then :
    The center of gravity of all Rational numbers is 1.5

    Indeed, it is very important to me to find any previous works on this matter.


    Arithmetic Mean of all rational numbers = 3/2
    Harmonic Mean of all rational numbers = 2/3
    Geometric Mean of all rational numbers = 1


    Many thanks,
    Domingo Gomez Morin
     
  18. May 16, 2010 #17
    Has anyone checked that the arithmetic mean of all rational numbers is 3/2 ?

    Am I wrong?

    If I am not wrong then try to imagine this: An infinite number of point-masses located
    in the real line at the same position of all the Rational-Numbers, then, when computing
    their Center of Mass it yields: 3/2.

    ¿?
    got it?

    How can this be physically posible?
    Are there the same number of point-masses within (0, 3/2) than within (3/2, Infinite)?

    How can this be mathematically posible?
    Are there the same number of rational numbers within (0, 3/2) than within (3/2, Infinite)?

    Why is 3/2 the equilibrium point of all rational numbers?

    Regards,
    Domingo Gomez Morin
     
    Last edited: May 16, 2010
  19. May 16, 2010 #18

    CRGreathouse

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    It's ill-defined. Depending on what order you choose, you can get a divergent series or convergence to any value desired.
     
  20. May 16, 2010 #19
    Yes, it might be ill-defined but what exactly is ill-defined here, let's see,
    The Stern-Brocot Tree is supposed to bring out all rational numbers in their reduced forms, So at an infinite stage I will have all rational numbers, isn't it?
    Now, when computing the arithmetic mean of all the Stern-Brocot fractions at any stage it
    always yields approximations closer and closer to 3/2, indeed, no matter what stage of the
    Stern-Brocot Tree you are considering.

    If it is ill-defined to say that the arithmetic mean of all rational numbers is 3/2, then
    it is ill-defined to say that the Stern-Brocot Tree brings out ALL the rational numbers.

    I understand that I have chosen the "Stern-Brocot Order", however this also means
    that it is also ill-defined to say that the Stern-Brocot Tree yields all the rational numbers.

    not exactly but kind of Zeno's paradoxe...
     
    Last edited: May 16, 2010
  21. May 16, 2010 #20

    CRGreathouse

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    It does generate all the rationals, yes.

    I trust you on this point, I'm not going to verify it tonight.

    False.

    False.

    How so?
     
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